Math
Continuity.
Definition
A function is continuous at a point x = a if three conditions are met: f(a) is defined, the limit as x approaches a exists, and the limit equals f(a). Informally, you can draw the function at that point without lifting your pencil.
Examples
Real-world.
- 1 f(x) = x² is continuous everywhere
- 2 f(x) = 1/x is not continuous at x = 0 because f(0) is undefined
- 3 A piecewise function may be discontinuous where the pieces meet if the values don't match
Key Fact
Continuous at a: (1) f(a) exists, (2) lim x→a f(x) exists, (3) lim x→a f(x) = f(a)